Combination Calculator

Calculate combinations using the formula nCr = n! / (r! × (n - r)!). Enter the total number of objects (n) and the number selected (r) to find how many different selections are possible when order does not matter. Get an exact answer and a step-by-step explanation.

Calculate nCr

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What Does nCr Mean?

nCr = n! / (r! × (n - r)!)

A combination counts the different ways to select r objects from n distinct objects when the order of selection does not matter.

Example: Choosing Alice and Bob is the same selection as choosing Bob and Alice. They count as one combination.

Enter n and r above to calculate the number of possible selections.

Combination Formula

The standard formula for selecting r objects from n distinct objects without regard to order is:

nCr = n! / (r! × (n - r)!)

The exclamation mark means factorial. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. The denominator removes the arrangements that differ only in the order of the selected objects.

n: total number of distinct objects.
r: number of objects selected.
nCr: number of unique selections.

The result is also called a binomial coefficient and is commonly written as C(n,r) or with n above r in parentheses.

Example: Calculate 5C3

Suppose there are five people and you want to select three people for a team. The order in which they are selected does not create a new team.

5C3 = 5! / (3! × (5 - 3)!)

= 120 / (6 × 2)

= 120 / 12

= 10

There are 10 different teams of three that can be selected from five people.

Combinations vs. Permutations

The key difference is whether order matters.

Combinations: order does not matter. Selecting three people for a team is an example.
Permutations: order matters. Assigning first, second, and third places in a race is an example.

Combinations: nCr = n! / (r! × (n - r)!)

Permutations: nPr = n! / (n - r)!

Use the Permutation Calculator when each different order counts as a separate outcome.

Important Properties of Combinations

Choosing none: nC0 = 1.
Choosing everything: nCn = 1.
Symmetry: nCr = nC(n - r).
Choosing one: nC1 = n, for n at least 1.

Symmetry means that choosing r objects to include is equivalent in count to choosing n - r objects to leave out. This property also helps make calculations more efficient.

Where Are Combinations Used?

Teams and committees: counting different groups selected from a larger set.
Probability: counting possible hands, samples, and selections.
Statistics: studying subsets and sampling outcomes.
Lottery mathematics: counting possible selections of numbers.
Computer science: working with subsets, combinatorial algorithms, and binomial coefficients.

Frequently Asked Questions

1. What is a combination?

A combination is a selection of objects in which the order of selection does not matter.

2. What is the formula for nCr?

The formula is nCr = n! / (r! × (n - r)!). It counts selections of r objects from n distinct objects.

3. What is 5C3?

5C3 = 5! / (3! × 2!) = 10. There are 10 ways to select three objects from five.

4. Can r be greater than n?

No. The number selected cannot exceed the total number available when selecting without replacement.

5. Why is nCr equal to nC(n - r)?

Selecting r objects to include leaves n - r objects out. Each selection corresponds uniquely to its complementary selection, so both counts are equal.

6. What is nC0?

nC0 = 1. There is exactly one way to select no objects: select the empty set.

7. Are the calculator results exact?

Yes. The calculator uses decimal string arithmetic and exact integer division instead of floating-point approximations. The input n is limited to 2000 to keep calculations and output manageable.